Digraphs Digraph Properties Digraph Applicationcs9024/11s2/classes/digraphs.pdf · for all e...

© 2010 Goodrich, Tamassia Directed Graphs 1

Directed Graphs

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Digraphs

A digraph is a graph whose edges are all directed Short for “directed graph”

Applications

one-way streets

flights

task schedulingA

C

E

B

D


Digraph Properties

A graph G=(V,E) such that

Each edge goes in one direction:

Edge (a,b) goes from a to b, but not b to a

If G is simple, m < n(n - 1)

If we keep in-edges and out-edges in separate adjacency lists, we can perform listing of incoming edges and outgoing edges in time proportional to their size

A

C

E

B

D


Digraph Application Scheduling: edge (a,b) means task a must be

completed before b can be started

The good life

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Directed DFS

We can specialize the traversal algorithms (DFS and BFS) to digraphs by traversing edges only along their direction

In the directed DFS algorithm, we have four types of edges

discovery edges

back edges

forward edges

cross edges

A directed DFS starting at a vertex s determines the vertices reachable from s

A

C

E

B

D


Reachability

DFS tree rooted at v: vertices reachable from v via directed paths

A

C

E

B

D

FA

C

E D

A

C

E

B

D

F


Strong Connectivity

Each vertex can reach all other vertices

a

d

c

b

e

f

g


Pick a vertex v in G

Perform a DFS from v in G

If there’s a w not visited, print “no”

Let G’ be G with edges reversed

Perform a DFS from v in G’

If there’s a w not visited, print “no”

Else, print “yes”

Running time: O(n+m)

Strong Connectivity Algorithm

G:

G’:

a

d

c

b

e

f

g

a

d

c

b

e

f

g


Maximal subgraphs such that each vertex can reach all other vertices in the subgraph

Can also be done in O(n+m) time using DFS, but is more complicated (similar to biconnectivity).

Strongly Connected Components

{ a , c , g }

{ f , d , e , b }

a

d

c

b

e

f

g


Transitive Closure

Given a digraph G, the transitive closure of G is the digraph G* such that

G* has the same vertices as G

if G has a directed path from u to v (u v), G*has a directed edge from u to v

The transitive closure provides reachability information about a digraph

B

A

D

C

E

B

A

D

C

E

G

G*


Computing the Transitive Closure We can perform

DFS starting at each vertex

O(n(n+m))

If there's a way to get

from A to B and from

B to C, then there's a

way to get from A to C.

Alternatively ... Use dynamic programming: The Floyd-WarshallAlgorithm


Floyd-Warshall Transitive Closure Idea #1: Number the vertices 1, 2, …, n.

Idea #2: Consider paths that use only vertices numbered 1, 2, …, k, as intermediate vertices:

k

j

i

Uses only verticesnumbered 1,…,k-1

Uses only verticesnumbered 1,…,k-1

Uses only vertices numbered 1,…,k(add this edge if it’s not already in)


Floyd-Warshall’s Algorithm Number vertices v1 , …, vn

Compute digraphs G0, …, Gn

G0=G

Gk has directed edge (vi, vj)

if G has a directed path from vi to vj with

intermediate vertices in {v1 , …, vk}

We have that Gn = G*

In phase k, digraph Gk is computed from Gk - 1

Running time: O(n3), assuming areAdjacent is O(1)

(e.g., adjacency matrix)

Algorithm FloydWarshall(G)

Input digraph G

Output transitive closure G* of G

i 1

for all v G.vertices()

denote v as vi

i i + 1

G0 G

for k 1 to n do

Gk Gk - 1

for i 1 to n (i k) do

for j 1 to n (j i, k) do

if Gk - 1.areAdjacent(vi, vk) Gk - 1.areAdjacent(vk, vj)

if Gk.areAdjacent(vi, vj)

Gk.insertDirectedEdge(vi, vj , k)

return Gn


Floyd-Warshall Example

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Floyd-Warshall, Iteration 1

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Floyd-Warshall, Conclusion

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DAGs and Topological Ordering

A directed acyclic graph (DAG) is a digraph that has no directed cycles

A topological ordering of a digraph is a numbering

v1 , …, vn

of the vertices such that for every edge (vi , vj), we have i < j

Example: in a task scheduling digraph, a topological ordering a task sequence that satisfies the precedence constraints

Theorem

A digraph admits a topological ordering if and only if it is a DAG

B

A

D

C

E

DAG G

B

A

D

C

E

Topological ordering of G

v1

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v4 v5


write c.s. program

play

Topological Sorting

Number vertices, so that (u,v) in E implies u < v

wake up

eat

nap

study computer sci.

more c.s.

work out

sleep

dream about graphs

A typical student day1

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1011

bake cookies


Note: This algorithm is different than the one in the book

Running time: O(n + m)

Algorithm for Topological Sorting

Algorithm TopologicalSort(G)

H G // Temporary copy of G

n G.numVertices()

while H is not empty do

Let v be a vertex with no outgoing edges

Label v n

n n - 1

Remove v from H


Implementation with DFS Simulate the algorithm by

using depth-first search

O(n+m) time.

Algorithm topologicalDFS(G, v)

Input graph G and a start vertex v of G

Output labeling of the vertices of Gin the connected component of v

setLabel(v, VISITED)

for all e G.outEdges(v) { outgoing edges }

w opposite(v,e)

if getLabel(w) = UNEXPLORED

{ e is a discovery edge }

topologicalDFS(G, w)

else

{ e is a forward or cross edge }

Label v with topological number n

n n - 1

Algorithm topologicalDFS(G)

Input dag G

Output topological ordering of Gn G.numVertices()

for all u G.vertices()

setLabel(u, UNEXPLORED)

for all v G.vertices()

if getLabel(v) = UNEXPLORED

topologicalDFS(G, v)


Topological Sorting Example



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Digraphs Digraph Properties Digraph Applicationcs9024/11s2/classes/digraphs.pdf · for all e...

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